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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An analysis of the practical DPG method
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by J. Gopalakrishnan and W. Qiu PDF
Math. Comp. 83 (2014), 537-552 Request permission

Abstract:

We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a “trial-to-test” operator $T$, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator $T$ is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply $T$. In practical computations, $T$ is approximated using polynomials of some degree $r > p$ on each mesh element. We show that this approximation maintains optimal convergence rates, provided that $r\ge p+N$, where $N$ is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.
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Additional Information
  • J. Gopalakrishnan
  • Affiliation: Department of Mathematics, Portland State University, P.O. Box 751, Portland, Oregon 97207-0751
  • MR Author ID: 661361
  • Email: gjay@pdx.edu
  • W. Qiu
  • Affiliation: The Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
  • Address at time of publication: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong
  • MR Author ID: 845089
  • Email: weifeqiu@cityu.edu.hk
  • Received by editor(s): July 21, 2011
  • Received by editor(s) in revised form: May 23, 2012
  • Published electronically: May 31, 2013
  • Additional Notes: Corresponding author: Weifeng Qiu
    This work was partly supported by the NSF under grants DMS-1211635 and DMS-1014817. The authors gratefully acknowledge the collaboration opportunities provided by the IMA (Minneapolis) during their 2010-11 program
  • © Copyright 2013 American Mathematical Society
  • Journal: Math. Comp. 83 (2014), 537-552
  • MSC (2010): Primary 65N30, 65L12
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02721-4
  • MathSciNet review: 3143683